We study unsupervised generative modeling in terms of the optimal transport
(OT) problem between true (but unknown) data distribution $P_X$ and the latent
variable model distribution $P_G$. We show that the OT problem can be
equivalently written in terms of probabilistic encoders, which are constrained
to match the posterior and prior distributions over the latent space. When
relaxed, this constrained optimization problem leads to a penalized optimal
transport (POT) objective, which can be efficiently minimized using stochastic
gradient descent by sampling from $P_X$ and $P_G$. We show that POT for the
2-Wasserstein distance coincides with the objective heuristically employed in
adversarial auto-encoders (AAE) (Makhzani et al., 2016), which provides the
first theoretical justification for AAEs known to the authors. We also compare
POT to other popular techniques like variational auto-encoders (VAE) (Kingma
and Welling, 2014). Our theoretical results include (a) a better understanding
of the commonly observed blurriness of images generated by VAEs, and (b)
establishing duality between Wasserstein GAN (Arjovsky and Bottou, 2017) and
POT for the 1-Wasserstein distance.

本文以优化传输问题为基础，研究了无监督生成建模，并使用惩罚最优传输（POT）目标来有效地最小化该问题。作者发现，AAE 中的目标与 POT for the 2-Wasserstein distance 相同，并与其他流行技术如 VAE 进行了比较。作者的理论结果包括更好地理解 VAEs 生成图像的模糊度以及建立 Wasserstein GAN 和 POT for the 1-Wasserstein distance 之间的对偶关系。